1. Field of the Invention
The present invention relates to a method of simulation of two-phase flow in pipes and notably hydrocarbon flow in a petroleum pipeline.
2. Description of the Prior Art
U.S. Pat. No. 5,550,761 describes a method for modeling the flows of two-phase mixtures in pipes using partial differential equations system comprising equations of mass conservation for the liquid and gas phase of the mixture and of total momentum conservation of the mixture. These equations are written as follows: ##EQU1## where R.sub.G (resp. R.sub.L) is the volume fraction of gas (resp. liquid), V.sub.G (resp. V.sub.L) is the velocity of the gas (resp. liquid) phase, P is the pressure, .rho.=.rho..sub.G R.sub.G +.rho..sub.L R.sub.L is the average density of the mixture, T.sup.w represents data depending on unknowns which represents the wall friction term, g is the acceleration of gravity and .theta. is the angle formed by the pipe with respect to the horizontal.
Three types of closing equations are added thereto:
1) a "conservative" closing law expressed in the form: EQU R.sub.L +R.sub.G =1
2) a hydrodynamic closing law written in the form: EQU .PHI.(V.sub.M, x.sub.G, .GAMMA.(P,T), dV, x)=0
where:
V.sub.M is the average velocity of the two-phase mixture,
x.sub.G is the mass fraction of gas ##EQU2##
.GAMMA.(P,T) represents all of the physical properties of the fluids,
T is the temperature,
dV=V.sub.G -V.sub.L is the slip velocity between the two phases
3) thermodynamic closing laws.
The partial differential equations system (1) is written in a conservative vectorial form: ##EQU3## F et Q are expressed as a function of W by means of hydrodynamic and thermodynamic laws but without an analytical expression for F(W,x) and Q(W,x).
The Jacobian matrix of the system with W is written: ##EQU4##
The eigenvalues .lambda..sub.1, .lambda..sub.2 and .lambda..sub.3 of A(W,x) are real and satisfy .lambda..sub.1 (W)&lt;.lambda..sub.2 (W)&lt;.lambda..sub.3 (W). They meet .lambda..sub.1 (W)&lt;0, .lambda..sub.3 (W)&gt;.sub.0. Furthermore, .lambda..sub.1 and .lambda.3 are greater than .lambda..sub.2 by at least one order of magnitude.
Modeling two-phase flows requires solving a relatively complex non linear hyperbolic partial differential equation system. It is well-known to use either an explicit or an implicit methodology to solve this type of non linear hyperbolic system. The drawback of the implicit methodology is poor accuracy. An explicit methodology produces more accurate results but, requires a time interval for solution that is difficult to use for two-phase flow in pipes because of particular constraints.
As mentioned above, the system of equations represents characteristics of having eigenvalues whose orders of magnitude are very different. Eigenvalues .lambda..sub.1 and .lambda..sub.3 are related to the velocities of the pressure waves in the pipe, which are of the order of several ten to several hundred m/s, thus much higher than the velocity of the fluids which is of the order of several m/s. The other eigenvalue, .lambda..sub.2, is related to the propagation velocity of the gas fraction in the pipe. With an explicit scheme, the time interval is determined by the greatest eigenvalue and it is thus relatively short with the present application.
This constraint is not compatible with an explicit methodology if the time interval value is within a reasonable calculating time range. One solution to avoiding this difficulty, is well-known and adopts an implicit solution methodology allowing selection of a greater time interval. On the other hand, the implicit methodology introduces a numerical diffusion which makes the results obtained less accurate, notably for slow waves.